Spectral analysis in a thin domain with periodically oscillating characteristics
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 427-451.

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.

DOI : 10.1051/cocv/2011100
Classification : 35P20, 49R05, 47A75, 35B27, 81Q10
Mots-clés : spectral analysis, dimension reduction, periodic homogenization, Γ-convergence, asymptotic expansions 
@article{COCV_2012__18_2_427_0,
     author = {Ferreira, Rita and Mascarenhas, Lu{\'\i}sa M. and Piatnitski, Andrey},
     title = {Spectral analysis in a thin domain with periodically oscillating characteristics},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {427--451},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {2},
     year = {2012},
     doi = {10.1051/cocv/2011100},
     mrnumber = {2954633},
     zbl = {1248.35135},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2011100/}
}
TY  - JOUR
AU  - Ferreira, Rita
AU  - Mascarenhas, Luísa M.
AU  - Piatnitski, Andrey
TI  - Spectral analysis in a thin domain with periodically oscillating characteristics
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 427
EP  - 451
VL  - 18
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2011100/
DO  - 10.1051/cocv/2011100
LA  - en
ID  - COCV_2012__18_2_427_0
ER  - 
%0 Journal Article
%A Ferreira, Rita
%A Mascarenhas, Luísa M.
%A Piatnitski, Andrey
%T Spectral analysis in a thin domain with periodically oscillating characteristics
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 427-451
%V 18
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2011100/
%R 10.1051/cocv/2011100
%G en
%F COCV_2012__18_2_427_0
Ferreira, Rita; Mascarenhas, Luísa M.; Piatnitski, Andrey. Spectral analysis in a thin domain with periodically oscillating characteristics. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 427-451. doi : 10.1051/cocv/2011100. http://archive.numdam.org/articles/10.1051/cocv/2011100/

[1] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | MR | Zbl

[2] G. Allaire and F. Malige, Analyse asymptotique spectrale d'un problème de diffusion neutronique. C. R. Acad. Sci. Paris, Ser. I 324 (1997) 939-944. | MR | Zbl

[3] H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984). | MR | Zbl

[4] N.S. Bachvalov and G.P. Panasenko, Homogenization of Processes in Periodic Media. Nauka, Moscow (1984). | MR | Zbl

[5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl

[6] G. Bouchitté, M.L. Mascarenhas and L. Trabucho, On the curvature and torsion effects in one dimensional waveguides. ESAIM : COCV 13 (2007) 793-808. | Numdam | MR | Zbl

[7] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston (1993). | MR | Zbl

[8] R. Ferreira and M.L. Mascarenhas, Waves in a thin and periodically oscillating medium. C. R. Math. Acad. Sci. Paris, Ser. I 346 (2008) 579-584. | MR | Zbl

[9] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, New York (1977). | MR | Zbl

[10] V. Jikov, S. Kozlov and O. Oleĭnik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994). | MR | Zbl

[11] S. Kesavan, Homogenization of Elliptic Eigenvalue Problems : Part 1. Appl. Math. Optim. 5 (1979) 153-167. | MR | Zbl

[12] S. Kesavan, Homogenization of Elliptic Eigenvalue Problems : Part 2. Appl. Math. Optim. 5 (1979) 197-216. | Zbl

[13] S. Kozlov and A. Piatnitski, Effective diffusion for a parabolic operator with periodic potential. SIAM J. Appl. Math. 53 (1993) 401-418. | MR | Zbl

[14] S. Kozlov and A. Piatnitski, Degeneration of effective diffusion in the presence of periodic potential. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 571-587. | Numdam | MR | Zbl

[15] F. Murat and L. Tartar, H-Convergence, in Topics in the mathematical modelling of composite materials. Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston (1997). | MR | Zbl

[16] O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization. Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam (1992). | MR | Zbl

[17] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math Sci. 90 (1981) 239-271. | MR | Zbl

[18] M.I. Vishik and L.A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl. (2) 20 (1962) 239-364 [English translation]. | MR | Zbl

Cité par Sources :